Introduction to Econometrics
Instructor: J´an Palguta
Worksheet for week # 4
1. Suppose that you are a director of university, you are considering to cancel the entrance
exams and you search for new criteria how to select good students for your
school. Assume you consider to use as the prior criteria the grades from high school
(in US called ACT scores).
In order to decide, some data about current students seem to be helpful. Let’s say
you have a sample of 15 students. The following table contains the ACT scores and
the GPA (grade point average) for these 15 students. Grade point average represents
the performance of the students at the university, it is based on a four-point scale
and has been rounded to one digit after the decimal.
Student GPA ACT
1 2.8 21
2 3.4 24
3 3.0 26
4 3.5 27
5 3.6 29
6 3.0 25
7 2.7 25
8 3.7 30
9 3.2 23
10 3.0 28
11 3.5 30
12 2.5 20
13 3.8 32
14 2.6 26
15 2.7 23
(a) Estimate the relationship between GPA and ACT using OLS; that is, obtain
the intercept and slope estimates in the equation GPA = β0 + β1 ACT. Use
Excel for the computation.
i. Compute intercept and slope coefficients using the summation formulas for
β0 and β1.
ii. Compute intercept and slope coefficients using matrix formula for β.1
(b) Comment on the direction of the relationship. Does the intercept have a useful
interpretation here? Explain. How much higher is the GPA predicted to be, if
the ACT score is increased by 5 points?
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You may want to check out the Excel commands =MMULT() and =MINVERSE().
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(c) Find and list the fitted values and the residuals of the model.
(d) Verify that the residuals (approximately) sum to zero.
(e) What is the predicted value of GPA when ACT=20?
2. This exercise serves to illustrate the distinction between the stochastic error term and
the residual. Usually, we can never observe the error term, but we can get around
this difficulty if we assume values for the true coefficients. Calculate values of the
error term and residual for each of the following six observations given that the true
β0 equals 0, the true β1 equals 1.5.
Yi 2 6 3 8 5 4
Xi 1 4 2 5 3 4
3. Open the dataset 401ksubs in Gretl located in (File → Open data → Sample File).
(a) Display the dataset for visual inspection. For this select all variable by pressing
Ctrl + A, then go to Data → Display values. Explore the following commands
in Data tab: Edit values, Add observations and Dataset structure.
(b) Plot the histogram of annual income (right click on inc, then choose Frequency
distribution).
(c) Compute descriptive statistics (right click on inc, then Summary statistics).
(d) Compute Gini coefficient for the income data (Variable → Gini coefficient).
Interpret your findings.
(e) Plot the histogram of ln(income). Explain how and why it is different from the
histogram of income.
4. Now split the sample into several sub-samples.
(a) Compute descriptive statistics and plot histograms for males and females (Sample
→ Restrict, based on criterion...). Interpret your findings.
(b) Re-do the above exercise for married and unmarried individuals. Explain your
intuition.
5. Let’s now move to bivariate graphs. Make three scatter plots: income against family
size, income against age and financial assets against income. Further, compute
correlation coefficients for each pair of variables. Explain your intuition.
6. Now let’s add time dimension to our analysis. Open dataset greene5 1 and make
time series plots of variables that you might find interesting.
7. Open dataset greene14 1 and explore the panel structure of the dataset. Make panel
plots of variable Q by company (unit).
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